Question
Circle C is intersected by tangent lines GH and JH. The two tangent lines intersect each other at point H. Identify the measure of ∠GHJ if m∠GCJ is 134°.(1 point)Responses67°67°226°226°134°134°46°
Answers
GPT-4o mini
To solve for the measure of \(\angle GHJ\), we can use the properties of circles and tangent lines.
Given that \(m\angle GCJ = 134^\circ\), we know that \(\angle GCJ\) is the angle formed by the line segment \(GC\) (from point \(G\) to the center of the circle \(C\)) and the tangent line \(HJ\) at point \(J\).
Since tangent lines form a right angle with the radius at the point of tangency, we have:
\[
m\angle GCJ + m\angle GHJ = 180^\circ
\]
The reason is that \(\angle GCJ\) and \(\angle GHJ\) are supplementary angles when viewed from point \(H\).
Now we substitute \(m\angle GCJ\) into the equation:
\[
134^\circ + m\angle GHJ = 180^\circ
\]
To find \(m\angle GHJ\), subtract \(134^\circ\) from \(180^\circ\):
\[
m\angle GHJ = 180^\circ - 134^\circ = 46^\circ
\]
Therefore, the measure of \(\angle GHJ\) is \(46^\circ\).
The answer is \(46^\circ\).
Given that \(m\angle GCJ = 134^\circ\), we know that \(\angle GCJ\) is the angle formed by the line segment \(GC\) (from point \(G\) to the center of the circle \(C\)) and the tangent line \(HJ\) at point \(J\).
Since tangent lines form a right angle with the radius at the point of tangency, we have:
\[
m\angle GCJ + m\angle GHJ = 180^\circ
\]
The reason is that \(\angle GCJ\) and \(\angle GHJ\) are supplementary angles when viewed from point \(H\).
Now we substitute \(m\angle GCJ\) into the equation:
\[
134^\circ + m\angle GHJ = 180^\circ
\]
To find \(m\angle GHJ\), subtract \(134^\circ\) from \(180^\circ\):
\[
m\angle GHJ = 180^\circ - 134^\circ = 46^\circ
\]
Therefore, the measure of \(\angle GHJ\) is \(46^\circ\).
The answer is \(46^\circ\).